Journal of Algebraic Combinatorics, 21, 269–279, 2005
c 2005 Springer Science + Business Media, Inc. Manufactured in The Netherlands.
On Modular Standard Modules
of Association Schemes
AKIHIDE HANAKI
Department of Mathematical Sciences, Faculty of Science, Shinshu University,
Matsumoto 390-8621, Japan
hanaki@math.shinshu-u.ac.jp
MASAYOSHI YOSHIKAWA
yoshi@math.shinshu-u.ac.jp
Department of Mathematical Sciences, Graduate School of Science and Technology, Shinshu University,
Matsumoto 390-8621, Japan
Received November 3, 2003; Revised June 24, 2004; Accepted July 1, 2004
Abstract. We will determine the structure of the modular standard modules of association schemes
of class two. In the process, we will give the theoretical interpretation for the p-rank theory for strongly regular graphs, and understand the p-rank as the dimension of a submodule of the modular standard module. Considering the modular standard module, we can obtain the detailed classification more than the p-rank and the
parameters.
Keywords: association scheme, modular adjacency algebra, modular standard module, p-rank, strongly regular
graph
1.
Introduction
There are many examples of nonisomorphic association schemes such that adjacency algebras over the complex number field have the same structure constants (intersection numbers). The standard module over the complex number field is completely determined by
the structure constants. So their standard modules over the complex number field are also
isomorphic. In this case, their modular adjacency algebras are also isomorphic to each other.
But there exist cases where their modular standard modules are nonisomorphic. Here, the
modular standard module means the standard module over a positive characteristic field.
Therefore the structure of the modular standard module can play a role for the structure
theory of the association schemes.
In this paper, we will determine the structure of the modular standard modules of association schemes of class 2. We will use the p-rank theory of strongly regular graphs which are
studied by Brouwer, van Eijl [5], and Peeters [8]. We will give the theoretical interpretation
for the p-rank theory, and understand the p-rank as the dimension of a submodule of the
standard module. Considering the standard module, we can obtain the detailed classification
more than the p-rank and the parameters.
270
2.
HANAKI AND YOSHIKAWA
Definition and preliminaries
In this section, we assume X to be commutative. Let X = (X, {Ri }i=0,... ,d ) be an association
scheme, and Ai (i = 0, . . . , d) be its adjacency matrices. Put n = |X | and vi the valency of
Ai . Let us denote the adjacency algebra of X over a field L by LX and the standard module
by L X .
Let (K , R, F) be a splitting p-modular system for the adjacency algebra, and let (π ) be
the maximal ideal of R. We denote the image of the canonical epimorphism R → F by ∗
(For details about a p-modular system, see [7]).
Suppose that X is commutative. Then
RX/π RX ∼
= FX
and π RX ⊆ J (RX) [7, Theorem I.14.1], so idempotents of FX are liftable to idempotents
of RX [7, Theorem I.14.2]. Consider the primitive idempotents decomposition of 1 FX in
FX:
1 FX = f 0 + · · · + f s ∈ FX,
then we have the primitive idempotents decomposition of 1 RX in RX:
1 RX = e B0 + · · · e Bs ∈ RX
such that e∗Bi = f i . These decompositions yield the decomposition of algebras. We call this
e Bi (e∗Bi ) a block idempotent of RX(FX), and we write Bi = e Bi RX and Bi∗ = e∗Bi FX.
Let e0 , . . . ,
ed be the set of primitive
idempotents in K X. Then there is a partition
{0, . . . , d} = sj=0 T j such that e Bi = j∈Ti e j . When e j ∈ Ti , we say that e j belongs to
the block Bi .
Let χ j be the (one-dimensional) irreducible representation of K X corresponding to e j .
Then e j belongs to Bi if and only if χ j (e Bi ) = 1. Since Bi∗ has the unique idempotent e∗Bi ,
Bi∗ /J (Bi∗ ) ∼
= F. If χi and χ j belong to the same block, then χi∗ = χ ∗j . So we have the
following.
Lemma 1 Irreducible representations χi and χ j of K X belong to the same block if and
only if χi (Ar ) ≡ χ j (Ar ) (mod (π)) for all r = 0, . . . , d.
Lemma 2 The dimension of Bi∗ is equal to the number of χ j belonging to Bi .
Proof:
We have dim F Bi∗ = rank R Bi = dim K e Bi K X.
We consider a (d + 1) × (d + 1) matrix P with the (i, j)-entry χi (A j ). We call P the
character table of X.
Proposition 3 The algebra FX is semisimple if and only if det P ≡ 0 (mod (π)).
MODULAR STANDARD MODULES OF ASSOCIATION SCHEMES
271
Proof: If FX is semisimple, then clearly we have det P ∗ = 0. If FX is not semisimple,
the P ∗ contains at least two same rows, so det P ∗ = 0.
General sufficient criteria for adjacency algebras to be semisimple were obtained in [1]
and [6].
Mapping each matrix Ar to vr induces an algebra homomorphism from K X to K . This
representation is usually called the trivial representation of K X. We assume that χ0 is the
trivial representation, and χ0 belongs to B0 . We call B0 the principal block of X. Then we
have the following.
Proposition 4 The dimension of the principal block B0∗ is one if and only if p n. (This
is also true for non-commutative association schemes.)
d
Proof: Put J = i=0
Ai . Since J Ai = Ai J = vi J , J ∗ generates a one-dimensional
ideal of FX, and the corresponding representation is trivial. We have J 2 = n J . If p n,
then n −1 J is a central primitive idempotent, thus B0∗ = J ∗ FX is one-dimensional. If p | n,
then J ∗ is a central nilpotent element in B0∗ , so it is in the Jacobson radical. In this case,
dim F B0∗ > 1.
3.
Contragradient modules
Let X = (X, {Ri }i=0,...,d ) be an association scheme (not necessary commutative), and F
a field. For a right FX-module V , we put V̂ := Hom F (V, F). For f ∈ V̂ , v ∈ V , and
i ∈ {0, . . . , d}, we define the action of FX to V̂ by
( f Ai )(v) := f (v Ai ).
Here Ai = t Ai . Then
( f (Ai A j ))(v) = f (v A j Ai ) = (( f Ai )A j )(v)
for i, j ∈ {0, . . . , d}.
So V̂ is a right FX-module. We call V̂ the contragradient module of V .
Proposition 5
F
X∼
= F X as right FX-modules.
Proof: For x ∈ X , define f x ∈ F
X by f x (y) = δx y . Put ϕ : F X → F
X the F-linear
map defined by ϕ(x) = f x . Obviously ϕ is an isomorphism of vector spaces. We show that
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HANAKI AND YOSHIKAWA
ϕ is an FX-homomorphism. We have
ϕ(x Ai )(y) = ϕ
z (y) =
f z (y)
=
(x,z)∈Ri
(x,z)∈Ri
1
if (x, y) ∈ Ri ,
0
otherwise,
(ϕ(x)Ai )(y) = ϕ(x)(y Ai ) = f x
=
z
(y,z)∈Ri 1
if (y, x) ∈ Ri ,
0
otherwise.
Now ϕ(x Ai ) = ϕ(x)Ai and ϕ is an FX-homomorphism.
Proposition 6 We fix x ∈ X . Put ι : FX → F X the F-linear map defined by ι(Ai ) = x Ai .
Then ι is an FX-monomorphism.
d
Proof: Obviously, ι is an FX-homomorphism. Assume ι( i=0
ai Ai ) = 0. Then
0=ι
d
ai Ai
=
i=0
d
ai x Ai =
d
i=0
ai y.
i=0 (x,y)∈Ri
Now ai = 0 for all i ∈ {0, . . . , d}, and so
d
i=0
ai Ai = 0.
We consider the structure of the contragradient module FX of the regular FX-module.
FX by τi (A j ) = δi j . Then {τi | i ∈ {0, . . . , d}} is a basis of FX. Now
Define τi ∈ (τi A j )(Ak ) = τi (Ak A j ) = pki j .
So we have the following.
Proposition 7 τi A j = dk∈0 pki j τk .
Proposition 8 Let ι : FX → F X be as above. Put ι̂ : F
X → FX the map defined by
ι̂( f ) = f ◦ ι. Then ι̂ is a FX-epimorphism.
Proof:
For i ∈ {0, . . . , d}, we fix y ∈ X such that (x, y) ∈ Ri . Then
ι̂( f y )(A j ) = f y ◦ ι(A j ) = f y (x A j )
1 if (x, y) ∈ R j ,
=
0 otherwise,
and so we have ι̂( f y ) = τi .
MODULAR STANDARD MODULES OF ASSOCIATION SCHEMES
4.
273
The structure of the adjacency algebras with d = 2
It is well-known that association schemes with at most five elements are commutative [12,
Theorem 4.5.1(ii)]. We assume that F is an algebraically closed field of characteristic p.
Lemma 9 The isomorphism classes of F-algebras of dimension 3 are F ⊕ F ⊕ F,
F ⊕ F[x]/(x 2 ), F[x]/(x 3 ), F[x, y]/(x 2 , x y, y 2 ), and T2 (F) = ( FF F0 ).
Since T2 (F) is non-commutative, T2 (F) can not be an adjacency algebra of an association
scheme. The algebra F[x, y]/(x 2 , x y, y 2 ) is not self injective, but it can be an adjacency
algebra of an association scheme. The smallest such example is the adjacency algebra of
the group association scheme of the symmetric group of degree 3 in characteristic 3.
Firstly, we consider the non-symmetric case, namely A1 = t A1 = A2 . Put k = v1 =
v2 = (n − 1)/2, where n = |X |. Note that k ≡ 3 (mod 4), in this case. Then we have
k−1
k+1
A1 +
A2 ,
2
2
k−1
k−1
A1 A2 = k A0 +
A1 +
A2 ,
2
2
k+1
k−1
A22 =
A1 +
A2 .
2
2
A12 =
Theorem 10 Let X be a non-symmetric association scheme with d = 2. Then we have
the following.
(1) If p n, then FX ∼
= F ⊕ F ⊕ F.
(2) If p | n, then FX ∼
= F[x]/(x 3 ).
Proof:
The character table is as follows:
A0
χ0
1
χ1
1
χ2
1
A1
k
√
−1+ n i
2
√
−1− n i
2
A2
k
√
−1− n i
2
√
−1+ n i
2
Obviously, (1) holds. (2) In this case, (A∗1 − k A∗0 )2 = 0 and (A∗1 − k A∗0 )3 = 0, so we have
FX ∼
= F[x]/(x 3 ).
Now we consider symmetric association schemes with d = 2. They correspond to
strongly regular graphs. If we consider a strongly regular graph with parameters (n, k, λ, µ)
[4, p. 8], then we have
A12 = k A0 + λA1 + µA2 ,
A1 A2 = (k − λ − 1)A1 + (k − µ)A2 ,
A22 = (n − k − 1)A0 + (n − 2k + λ)A1 + (n − 2k + µ − 2)A2 .
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HANAKI AND YOSHIKAWA
Now A1 has three eigenvalues k, r , and s, here r and s are roots of the equation x 2 +
(µ − λ)x + (µ − k) = 0. The character table is as follows.
A0
A1
A2
χ0
1
k
n−k−1
χ1
1
r
−r − 1
χ2
1
s
−s − 1
If the graph is not a conference graph, then r and s are rational integers.
Theorem 11 Let be a strongly regular graph with parameters (n, k, λ, µ) which is
not a conference graph, and let X be an association scheme defined by . Then A1 has
eigenvalues k, r , and s and the following holds.
(1) If p | n and r ≡ s (mod p), then FX ∼
= F[x]/(x 2 ) ⊕ F (F[x]/(x 2 ) is the principal
block),
(2) If p | n, r ≡ s (mod p), and at least one of k(k + 1), λ − 2k and µ are not zero (mod
p), then FX ∼
= F[x]/(x 3 ),
(3) If p | n, r ≡ s (mod p), and k(k + 1) ≡ λ − 2k ≡ µ ≡ 0 (mod p), then FX ∼
=
F[x, y]/(x 2 , x y, y 2 ),
(4) If p n and r ≡ s (mod p), then FX ∼
= F ⊕ F ⊕ F,
(5) If p n and r ≡ s (mod p), then FX ∼
= F ⊕ F[x]/(x 2 ) (F is the principal block).
In (2) and (3), k(k + 1) ≡ λ − 2k ≡ µ ≡ 0 (mod p) if and only if (A∗1 − k A∗0 )2 = 0.
Proof: We note that r ≡ s (mod (π )) if and only if r ≡ s (mod p) since r and s are
rational integers.
Suppose that p | n. Then the principal block is not simple, so k ≡ r (mod p) of k ≡ s
(mod p) by the character table. If only one of them holds, then (1) holds. Otherwise,
FX has only one irreducible representation, so it is a local algebra. In this case, we have
FX ∼
= F[x]/(x 3 ) or FX ∼
= F[x, y]/(x 2 , x y, y 2 ). If (A∗1 − k A∗0 )2 = 0, then we have
3
∼
FX = F[x]/(x ). Suppose (A∗1 − k A∗0 )2 = 0. Then, by a direct calculation, we have
k(k + 1) ≡ λ − 2k ≡ µ ≡ 0 (mod p) and (α A∗1 + β A∗2 )2 = 0 for any α, β ∈ F. Thus
FX ∼
= F[x, y]/(x 2 , x y, y 2 ).
Suppose that p n. Then the principal block is isomorphic to F. The result follows from
Lemma 2.
Note that there exist strongly regular graphs for every case in this theorem.
Now we consider conference graphs. Then the parameters
are (n, k, λ, µ) √
= (4µ +
√
1, 2µ, µ − 1, µ), and the eigenvalues are k, r = (−1 + n)/2, and s = (−1 − n)/2.
Theorem 12 Let be a conference graph with parameters (n, k, λ, µ) = (4µ+1, 2µ, µ−
1, µ), and let X be an association scheme defined by . Then the following holds.
(1) If p | n, then FX ∼
= F[x]/(x 3 ).
(2) If p n, then FX ∼
= F ⊕ F ⊕ F.
MODULAR STANDARD MODULES OF ASSOCIATION SCHEMES
275
Proof: Suppose p | n. Then k ≡ r ≡ s (mod (π )), so FX is not semisimple. We can
easily check that (A∗1 − k A∗0 )2 =√0, thus FX ∼
= F[x]/(x 3 ).
Suppose p n. Then r − s = n ≡ 0 (mod (π )). Thus FX is semisimple.
5.
The structure of the standard module with d = 2
In the previous section, we showed that there exist the following five types for the structure
of the adjacency algebras:
(1)
(2)
(3)
(4)
(5)
FX ∼
=
FX ∼
=
FX ∼
=
FX ∼
=
FX ∼
=
F ⊕ F ⊕ F,
F ⊕ F[x]/(x 2 ) (the principal block is F),
F[x]/(x 2 ) ⊕ F (the principal block is F[x]/(x 2 )),
F[x]/(x 3 ),
F[x, y]/(x 2 , x y, y 2 ).
In this section, we will determine the structure of the standard module for each type. We
assume that A1 has eigenvalues k, r, s with multiplicities 1, m r , m s .
If X is commutative, (F X )M is an FX-module for any M ∈ FX. Then it follows that
dim(F X )M = r k F (M). Especially, if X is class 2, the p-rank of M such that M ∈ radFX
is very important. Since many of them have the relevant p-rank, the structure of F X plays
a role for the structure theory of the association scheme (See [5]).
5.1.
Type (1) FX ∼
=F⊕F⊕F
Let B0 , B1 and B2 be the corresponding blocks to k, r and s, respectively. Let Si be the
simple module such that the corresponding representation belongs to Bi for i = 0, 1, 2. In
this case, it is known that
FX ∼
= S0 ⊕ m r S1 ⊕ m s S2 .
In this case, the structure of F X is completely determined by the parameters.
In the case of type (2) or type (3), we set that FX = B0 ⊕ B1 is the block decomposition
of the adjacency algebra, where B0 is the principal block. Let Si be the simple module such
that the corresponding representation belongs to Bi for i = 0, 1. In these cases, it is enough
that we determine dim F radF X .
5.2.
Type (2) FX ∼
= F ⊕ F[x]/(x 2 )
Let P be the projective cover of S1 and e1 the block idempotent of B1 . We set that M =
e1 (A1 − s A0 ). Then radFX = F M ∗ . Since the coefficient of Ai∗ in M ∗ is in F p for each i,
r k F (M) = r k p (M). We set that t = r k p (M). Then it follows that
FX ∼
= S0 ⊕ t P ⊕ (n − 1 − 2t)S1 .
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HANAKI AND YOSHIKAWA
Interesting examples are the association schemes defined by (26,10,3,4)-strongly regular
graphs. In this case, there are 10 non-isomorphic association schemes. The adjacency algebras over a field of characteristic 5 are type (2). Then the structures of the modular standard
modules are as follows:
FX
5.3.
The number of non-isomorphic
association schemes
S0 ⊕ 9P ⊕ 7S1
1
S0 ⊕ 11P ⊕ 3S1
2
S0 ⊕ 12P ⊕ S1
7
Type (3) FX ∼
= F[x]/(x 2 ) ⊕ F
Let P be the projective cover of S0 . In this case, we have either k ≡ r or k ≡ s. Without
loss of generality, we assume that k ≡ r . Since radFX is F J ∗ , radF X is one-dimensional.
Therefore it follows that
FX ∼
= P ⊕ (n − m s − 2)S0 ⊕ m s S1 .
In this case, the structure of F X is completely determined by the parameters.
5.4.
Type (4) FX ∼
= F[x]/(x 3 )
We set that Mi ∼
= FX/(radFX)i as FX-modules for i = 1, 2, 3. We set M = A1 − k A0 .
Then rad2 FX = F J ∗ and radFX = F J ∗ + F M ∗ .
We set t = r k p (M) = r k F (M). Since (M ∗ )2 = µJ ∗ = 0, (F X )J ∗ ⊂ (F X )M ∗ .
Therefore it follows that
FX ∼
= M3 ⊕ (t − 2)M2 ⊕ (n − 2t + 1)M1 .
Interesting examples are the association schemes defined by the (25,12,5,6)-strongly
regular graphs. Their adjacency algebras over a field of characteristic 5 are type (4). There
are 15 strongly regular graphs with parameters (25,12,5,6), but the only one of them is selfcomplementary, so there are 8 non-isomorphic association schemes. Then the structures of
the modular standard modules are as follows:
FX
The number of non-isomorphic
association schemes
M3 ⊕ 10M2 ⊕ 2M1
5
M3 ⊕ 9M2 ⊕ 4M1
2
M3 ⊕ 7M2 ⊕ 8M1
1
MODULAR STANDARD MODULES OF ASSOCIATION SCHEMES
5.5.
277
Type (5) FX ∼
= F[x, y]/(x 2 , x y, y 2 )
In this case, p | v1 and p | v2 + 1, or p | v1 + 1 and p | v2 . Let α and β be p | vα and
p | vβ + 1. In this case, we can define an isomorphism FX → F[x, y]/(x 2 , x y, y 2 ) by
Aα → x and A0 + Aα + Aβ → y. From here, we identify them.
Now we consider FX. Consider a basis {τ0 , τα − τ0 , τ0 − τβ } of FX. Then the matrix
representation of FX on FX with respect to this basis is as follows:

0

x →  0
0
0
0


0

1,
0
0
0

y →  0
0
0
1

0

0.
0
0
Now we consider the structure of F X . Let
F X = M1 ⊕ M2 ⊕ · · · ⊕ Mr
be an indecomposable decomposition of F X . Since dim F F X y = 1, there is the unique Mi
such that Mi y = 0. We assume M1 y = 0.
Since FX has tame representation type, we can classify all indecomposable FX-modules
[3, Section 4.3]. We have the following.
Proposition 13 Let M be an indecomposable FX-module such that dim F M y = 0. Then
the corresponding representation is one of the following:
1 : x → (0), y → (0) (simple module),
0 1
0 0
2 : x →
, y →
.
0 0
0 0
Proposition 14 Let M be an indecomposable FX-module such that dim F M y = 1. Then
the corresponding representation is one of the following:
1 (λ) : x →
2 : x →

0
0
0
0
0

3 : x →  0
0
1
,
0
0
,
0
0
0
λ
y →
(λ = 0),
0
0 1
y →
,
0 0



1
0 0 0



0  , y →  0 0 1  ,
0
0
0
0
0
0
0
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HANAKI AND YOSHIKAWA

0

4 : x →  0
0

0
0

5 : x → 
0
0
1
0


0
0

0,
0
0
1
0
0
0
0
0
0
0

0

y →  0
0
1

,
0
0
0

1

0
0

0,
0
0
0
0
0
0
0
0
0
0
0
0

y → 
0
1

0

.
0
0
We set t = r k p (Aα ) = r k F (Aα ). From Propositions 5 and 13, M1 must be selfcontragradient. It follows that F X ∼
= 5 ⊕ (t − 2)2 ⊕ (n − 2t)1 , 1 (λ) ⊕
(t − 1)2 ⊕ (n − 2t)1 , or 2 ⊕ t2 ⊕ (n − 2t − 2)1 .
Let us denote the column space of Aα by Aα and the all-one column vector by 1. Then
(F X )J ∗ ⊂ (F X )Aα if and only if 1 ∈ Aα . We
that 1 ∈ Aα . Let us denote the
assume
n
ai Aα (i), where ai ∈ F for all i. Then
i-th column vector of Aα by Aα (i). We set 1 = i=1
it follows that
FX ∼
=




5 ⊕ (t − 2)2 ⊕ (n − 2t)1




if 1 ∈ Aα and

1 (λ) ⊕ (t − 1)2 ⊕ (n − 2t)1 if 1 ∈ Aα and






2 ⊕ t2 ⊕ (n − 2t − 2)1
if 1 ∈ Aα .
n
i=1
n
ai ≡ 0 (mod p),
ai ≡ 0 (mod p),
i=1
The example whose modular standard module contains 5 is the association scheme
defined by the (16,5,0,2)-strongly regular gaph. The adjacency algebra over a field of characteristic 2 is type (5) and F X ∼
= 5 ⊕ 42 ⊕ 41 .
The example whose modular standard module contains 2 is the association scheme
defined by the triangular graph T (8). The adjacency algebra over a field of characteristic 2
is type (5) and F X ∼
= 2 ⊕ 62 ⊕141 . There are 4 strongly regular graphs with parameters
(28, 12, 6, 4), namely T (8) and three Chang graphs. We know that T (8) is characterized by
the 2-rank r k2 (A1 ) (See [5]), but T (8) is also characterized by the structure of the modular
standard module. Because it follows that F X ∼
= 5 ⊕ 62 ⊕ 121 for the association
scheme defined by a Chang graph.
The example whose modular standard module contains 1 (λ) is the association scheme
defined by the (36,14,4,6)-strongly regular graphs. The adjacency algebras over a field
of characteristic 3 are type (5). There are 180 strongly regular graphs with parameters
(36,14,4,6). Two of them have the structure of the modular standard module F X ∼
= 1 (λ) ⊕
122 ⊕ 101 .
Notes In the cases type (1) and (3), the structure of the modular standard module is determined completely by the multipicities (if the association scheme is defined by a strongly
regular graph, they are determined by the parameters). Therefore we are interested in type
MODULAR STANDARD MODULES OF ASSOCIATION SCHEMES
279
(2), (4) and (5). In the cases type (2) and (4), we can determine the structure of the modular standard module by the p-rank of the corresponding matrix and the parameters of the
corresponding strongly regular graph.
Here we focus on the case type (5). Then there is the possibility that we can obtain more
information than the p-rank. Namely, they have the same p-rank of Aα , but their structures of
the modular standard module are non-isomorphic. Such an example is (36, 15, 6, 6)-strongly
regular graph. There are 227 strongly regular graphs with the parameter (36, 15, 6, 6) in
the list by Spence [10], [11]. There are 60 strongly regular graphs such that r k2 (A2 ) = 14.
Then three strongly regular graphs of them have F X ∼
= 2 ⊕ 142 ⊕ 61 , and the others
FX ∼
= 5 ⊕ 122 ⊕ 81 . We do not know the examples that there are all of three cases
with the same p-rank.
In general, it is not so easy to calculate the value λ for 1 (λ). We do not know the value
λ and its range.
Acknowledgment
The authors are thankful to the referee for lots of helpful remarks and suggestions.
References
1. Z. Arad, E. Fisman, and M. Muzychuk, “Generalized table algebras,” Israel J. Math. 144 (1999), 29–60.
2. E. Bannai and T. Ito, “Algebraic combinatorics. I,” Association Schemes, Benjamin-Cummings, Menlo Park,
CA, 1984.
3. D. Benson, Representation and Cohomology I, Cambridge, 1995.
4. A.E. Brouwer, A.M. Cohen, and A. Neumaier, Distance-Regular Graphs, Springer-Verlag, Berlin, Heidelberg,
1989.
5. A.E. Brouwer and C.A. van Eijl, “On the p-rank of the adjacency matrices of strongly regular graphs,” J.
Algebraic Combin. 1 (1992), 329–346.
6. A. Hanaki, “Semisimplicity of adjacency algebras of association schemes,” J. Algebra 225 (2000), 124–129.
7. H. Nagao and Y. Tsushima, Representations of Finite Groups, Academic Press, San Diego CA, 1987.
8. R. Peeters, “Uniqueness of strongly regular graphs having minimal p-rank,” Linear Algebra Appl. 226–228
(1995), 9–31.
9. R. Peeters, “On the p-ranks of the adjacency matrices of distance-regular graphs,” J. Algebraic Combin. 15
(2002), 127–149.
10. E. Spence, “Regular two-graphs on 36 vertices,” Linear Algebra Appl. 459–497 (1995), 226–228.
11. E. Spence, http://www.maths.gla.ac.uk/%7ees/srgraphs.html .
12. P.-H. Zieschang, An Algebraic Approach to Association Schemes, vol. 1628, Lecture Notes in Math Springer,
Berlin, Heidelberg, New York, 1996.